UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Example Problems Example (1) In the problem below, find the demand curve for X and the Engel curve for X. max U = 2*ln(3X) X subject to: P x X ≤ I, where P x is the per-unit price of X, and I is income allocated to purchase of X. Note that both P X and I are parameters. Converting the problem into an equivalent Lagrangian problem, and introducing the new choice variable λ into the problem as a Lagrangian multiplier: max L = 2*ln(3X) + λ (I - P x X) X, λ subject to: nothing F.O.C.'s: (1)0 Px 3 ) X 3 1 ( 2 X L = λ ⋅-⋅ ⋅ = ∂ ∂ (2)0 X Px I L = ⋅-= λ ∂ ∂ The FOC equations provide us with two equations in two unknowns, X and λ . Solve these two equations for the solution values of X and λ . There are several ways to work the algebra to solve these two equations. One way is illustrated below: Solve FOC (2) for X: I - P x X = 0 I = P x X X * = I/P x (Note: we typically use a star "*" superscript to denote a solution value.) Note: We know that the equation above gives the solution for X because X is alone on one side of the equation, and everything else on the other side of the equation is a parameter. At this point, we have the demand curve for X. The demand curve for X is: X * = I/P x , where X * and P X are allowed to vary, and I is held constant. We also have the Engel curve for X.

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